The Compact Support Property of Rough Super Brownian Motion on $\mathbb{R}^2$

Abstract: We discuss the compact support property of the rough super-Brownian motion constructed as a scaling limit of a branching random walk in static random environment. The semi-linear equation corresponding to this measure-valued process is the continuous parabolic Anderson model, a singular SPDE in need of renormalization, which prevents the use of classical PDE arguments. But with the help of an interior estimation method, we are able to show that the compact support property also holds for rough super-Brownian motion.